Double electron capture in fast ion-atom collisions

Abstract

A four-body formalism of the modified target continuum distorted wave (MTCDW-4B) with incorrect boundary conditions and boundary corrected continuum intermediate state (BCCIS-4B) approximation have been employed to calculate the differential cross sections (DCS) and total cross sections (TCS) for double-electron capture (DC) in collision of fast bare ions with helium atoms in their ground states. In both these formalisms, the intermediate continuum state of each of the active electrons with the target has been taken into account. The influence of the static-electron-correlations on cross sections has also been taken into account by choosing the proper wave functions of the initial states of the bound electrons. Moreover, we have computed the cross sections using the asymptotic Coulomb logarithmic phase for the relative motion of two colliding nuclei in BCCIS-4B theory. The present computed results are compared with the available experimental and other existing theoretical results. TCS are found to be in good agreement with the measurements. In addition, we have also analyzed DCS for DC in the collision of $\alpha$ -particles with helium atoms at intermediate and high projectile energies. We have investigated the significance of the contributions to TCS and DCS from excited states (both single and double) of He and $\textit{Li}^{+}$ especially in comparison between theories and measurements. For symmetric collision, the ground state capture dominates over the excited states whereas for asymmetric collision, the excited states including singly and doubly excited states dominate over the ground state capture. It is also clear that the ground state contribution dominates at very high impact energies for asymmetric collision. The obtained results for DCS into the ground state and excited states are compared with the experimental data and overall satisfactory agreements have been found at different impact energies.

Keywords

Lon-atom collision electron capture cross sections

1. Introduction

In recent decades, experimental advancements have enabled accurate measurements at kinematically complete fully differential cross sections (FDCS). For three-body collisions, several dynamical aspects of the elementary processes such as electron excitation, ionisation and capture have been elucidated in more details, while for many-electron systems the most addressed question is the influence of electron correlation on the magnitude of the process [1, 2, 3]. However DC is a particularly interesting case of a two-electron process which is important for their practical applications in different fields of physics such as astrophysics, plasma physics and controlled thermonuclear fusion researches. Thus the details of the transfer of two electrons from a bound state of the target to a bound state of the projectile are sensitive to the static electron-electron correlation in the initial and final state, as well as to the dynamic correlation during the collision. Due to rapid expansion of computer facilities, increasing theoretical attention has been paid in the last twenty years to atomic collision involving two electron processes [4, 5, 6] in the intermediate to high energy region. For example in DC following energetic proton-helium collisions, the four-body formalism of continuum distorted-wave (CDW-4B) method, formulated in Ref. [4] and applied further, in Refs [5, 6], has been shown to predict very well the experimental data above 100 keV. In the present theoretical investigation, we are motivated to study the DC of helium atom by the impact of bare ion within the framework of the four-body modified target continuum distorted-wave (MTCDW-4B) with incorrect boundary conditions and boundary corrected continuum intermediate state (BCCIS-4B) approximation in the intermediate to high energies. It may be mentioned that BCCIS approximation has been introduced by Mandal et al. [7] to study the formation of positronium in $e^{+}+H$ collision so that the theory goes beyond first order. BCCIS-4B is actually the extension of BCCIS-3B theory to four-body collision. Similar development has been initiated from three-body target continuum distorted wave (TCDW-3B) [8] theory to MTCDW-4B.

Reviews of different quantum-mechanical four-body distorted wave theories for DC in collisions between fast heavy multiply charged ions and helium like atomic systems are given in Refs [6, 9, 10, 11]. The four-body boundary-corrected continuum-intermediate state (BCIS-4B) [12] and the four-body Born distorted wave (BDW-4B) [6] approximations have been introduced to study general DC from an arbitrary two-electron system by any heavy nuclei as projectile and illustrated for the ground-to-ground-state transition in the $\textit{He}^{2+}+\textit{He}$ collisions at 900–7000 keV. Subsequently [10], for the latter scattering system, the computations in the BCIS-4B and BDW-4B methods have covered the extended impact energy range 100–7000 keV. Crothers and McCarroll [13] have investigated the DC and single capture (SC) of electron from helium atom by fast alpha particles within a correlated version of the CDW approximation in the small energy ranges from 500 to 1400 keV. In this calculation, the independent electron model (IEM) have been avoided. The DC by fast bare ions in collisions with helium have been studied by Gayet et al. [14] within the CDW-4B approximation in the energy range of 0.4-30 MeV. In this work [14], the contributions of TCS from ground state, singly excited states and doubly excited states have also been analyzed. Following the lead from Refs [4, 5], it has been found in Ref. [16] that the TCS for DC from the CDW-4B method compare favorably with the experimental data. By contrast, the experimentally measured TCS for DC are grossly underestimated (within 2–4 orders of magnitude) by the four-body continuum distorted-wave eikonal initial state (CDW-EIS) [15] as further analyzed in Refs [9, 11]. It should be noted that in the CDW-4B, BCIS-4B, BCCIS-4B and MTCDW-4B methods both electrons jump simultaneously from the target into the projectile to form a two-electron atomic system in the exit channel. The DC cross sections for collisions of bare ions $(\textit{He}^{2+},$ $\textit{Li}^{3+}$ and $B^{5+})$ with helium atoms have been investigated at intermediate and high energies using BCCIS-4B approximation [16, 17, 18]. In the BCCIS method, the correct boundary conditions (CBC) are satisfied by the total scattering wave function, whereas the MTCDW method does not satisfy the CBC in neither the entrance nor the exit channels. In both formalisms, the continuum state of each electron has been taken. The importance of correct boundary conditions in ion-atom collision and for different formalisms have been thoroughly discussed in Refs [9, 11, 19]. In the calculations for DC treated in Refs [9, 11], the uncorrelated one-parameter hydrogen like wave functions with the effective charge $(z_{\textit{eff}}=$ 1.6875) for initial bound state of He atom have been taken. The results obtained by BCCIS-4B approximation [16, 17, 18] for TCS are in good agreement with the existing experimental data. However, Schulz et al. [20] have measured DCS for DC of $p-\textit{He}$ collisions in the projectile energies ranging from 15–150 keV. They have found that the experimental data exhibit the typical steeply decreasing dependence on the scattering angle, but neither the oscillating structures due to characteristic of interference effects nor the Thomas peak [21] for SC at large energies are observed [22, 23]. Later, theoretically the electron dynamics in $p-\textit{He}$ and $\textit{He}^{1,2+}-\textit{He}$ collisions have been investigated on the basis of IEM by using the two-center basis generator method (TC-BGM) [24] which neglects electron-electron correlation effects. This study shows surprisingly that electron correlations play minor role for the processes at impact energies in the range from 40 to 630 keV/amu. Using the cold target ion momentum spectroscopy technique, Schoffler et al. [25] have measured state-selective projectile scattering angles for electron transfer in collision of proton and dressed ions with helium atom for incident energies of 60–630 keV/amu. They have also reported theoretical results obtained by means of four-body one channel distorted-wave models called continuum distorted-wave Born final state (CDW-BFS) and continuum distorted-wave Born initial state (CDW-BIS) [26] that are, according to Refs [9, 11, 12, 27] alternatively called the prior and post forms of the four-body Born distorted-wave (BDW-4B) model, respectively. The agreement of these models with experimental data is good similar to that with the BCIS-4B model. Further, Harris et al. [28] have presented a four-body model for DC (4BDC) to study the FDCS for $p-\textit{He}$ collisions. In this model, the effects of initial and final state electron correlations have been studied as well as the role of the projectile-nucleus interaction. It was shown that correlation has minor effect on the FDCS. Projectile angular distributions of single and double electron transitions in collision of helium with energetic bare projectile impacts have been studied within the framework of Born and CDW approximation [29]. These calculations have been performed within the IEM where the correlated motion of the electrons are neglected. In this paper, multiple critical scattering in the projectile angular distribution have been observed which has been verified by the experiments of Gudmundsson et al. [30]. Recently, Ghavaminia and Ghanbari-Adivi [31] have calculated the DCS and the TCS for DC by protons from the ground state helium atoms at intermediate and high impact energies by using Coulomb Born distorted-wave approximation (CBDW). They have calculated the DC cross sections into the ground state only. This calculation shows that the initial state correlations play a minor role while the role of the final state correlations is fundamental, which causes considerable differences between different DCS results using different final wave functions. It is well known that any study on electron transfer whether single or double may find success only when intermediate continuum state(s) of the active electron(s) is(are) embodied into the formalism. In a four-body collision, six two-body interactions and three relative kinetic energy terms constitute the total hamiltonian. Depending on the splitting of the total hamiltonian, channel hamiltonians and interactions in respective channels (entrance or exit) are determined in such a way that Coulomb boundary conditions are satisfied accordingly. Next comes the determination of the approximate version of total wavefunction to be used in either form (post or prior) of the transition amplitude. Different combinations constitute the different approximate methods with the corresponding acronyms. Though the derivations of the transition amplitudes (in either form) for the BCCIS-4B and BCIS-4B approximations have been made in two different ways, the forms of the transition amplitudes differ only in the wavefunctions for the relative motion of two colliding heavy nuclei. For this latter motion, the BCIS approximation uses the logarithmic Coulomb phase wavefunction, whereas the BCCIS-4B approximation employs the full Coulomb wavefunction. Otherwise, both the full Coulomb wavefunction and its asymptotic form given by the mentioned logarithmic Coulomb phase are defined on the energy shell. In this paper, in continuation of our previous studies, we employ the BCCIS-4B and the MTCDW-4B [32] approximations to investigate the DC process in energetic bare ion-helium collisions. The correlated multi-parameter wave functions are used to explore the influence of the electronic correlation on the rearrangement process. DCS contributions for DC in $\textit{He}^{2+}-\textit{He}$ collisions from ground state and excited states to TCS have been investigated at different projectile energies. Comparisons are also made between the present results and the corresponding experimental findings as well as the results obtained from other theories.

The organization of this paper is as follows. In Section 2, the theoretical formalism is outlined for both BCCIS-4B and MTCDW-4B models. The obtained results for DCS and TCS and their related comparisons and discussion are given in Section 3. Finally, a summary is given in Section 4. Atomic units are used throughout unless stated otherwise.

2. Theory

A four-body collision is considered in which a bare charged projectile (P) ion of mass $M_{P}$ and charge $z_{P}$ is incident upon an two-electron neutral target (T) (helium atom) of nuclear mass $M_{T}$ and charge $z_{T}=$ 2; which is at rest in the laboratory frame. The electrons are denoted by $e_{1}$ and $e_{2}$ and their position vectors relative to the target (projectile) nucleus are denoted by $\vec{r}_{1T}$ and $\vec{r}_{2T}$ $(\vec{r}_{1P}$ and $\vec{r}_{2P})$ . We assume that the two electrons are initially bound to the target nucleus in their ground state and are captured directly into the ground or excited states at the projectile during the collision process. Let $\vec{R}$ denotes the position vector of the projectile relative to the target nucleus. $\vec{r}_{i}$ is the position vector of the projectile with respect to the center of mass of the target atom $(T;e_{1},e_{2})$ and $\vec{r}_{f}$ is that of the target ion with respect to the center of mass of the newly-formed projectile $(P;e_{1},e_{2})$ ion. Here the position vector of $e_{2}$ relative to $e_{1}$ is given by $\vec{r}_{12}$ , where $\vec{r}_{12}=\vec{r}_{2T}-\vec{r}_{1T}=\vec{r}_{2P}-\vec{r}_{1P}$ . The total Hamiltonian (H) of the whole system, in the center of mass coordinate system corresponding to the initial and the final channels respectively, may be written as,

$\displaystyle H=H_{0}-\frac{z_{T}}{r_{1T}}-\frac{z_{T}}{r_{2T}}+\frac{1}{r_{12% }}+V_{i}=H_{i}+V_{i}=H_{0}-\frac{z_{P}}{r_{1P}}-\frac{z_{P}}{r_{2P}}+\frac{1}{% r_{12}}+V_{f}=H_{f}+V_{{f}}$ (1)

where

$\displaystyle H_{0}=-\frac{1}{2\mu_{i}}\nabla^{2}_{r_{i}}-\frac{1}{2a}\nabla^{% 2}_{r_{1T}}-\frac{1}{2a}\nabla^{2}_{r_{2T}},=-\frac{1}{2\mu_{f}}\nabla^{2}_{r_% {f}}-\frac{1}{2b}\nabla^{2}_{r_{1P}}-\frac{1}{2b}\nabla^{2}_{r_{2P}},$

and

$\displaystyle V_{i}=\frac{z_{P}z_{T}}{R}-\frac{z_{P}}{r_{1P}}-\frac{z_{P}}{r_{% 2P}},V_{f}=\frac{z_{P}z_{T}}{R}-\frac{z_{T}}{r_{1T}}-\frac{z_{T}}{r_{2T}},$ (2)

where the reduced masses $\mu_{i},\mu_{f}$ , a, b are given by $\mu_{i}=M_{P}(M_{T}+2m_{e})/(M_{P}+M_{T}+2m_{e}),\mu_{f}=M_{T}(M_{P}+2m_{e})/(% M_{P}+M_{T}+2m_{e}),a=M_{T}m_{e}/(M_{T}+m_{e}),b=M_{P}m_{e}/(M_{P}+m_{e})$ . It is obvious that the perturbation potentials $V_{i}$ and $V_{f}$ vanish for large separation between the distinct fragments in entrance and exit channels for $\alpha$ -particle impact. There is no Coulomb distortion at infinity for initial and final bound state wave functions. The transition amplitude for a double electron capture process may be written as,

$\displaystyle T_{if}=\langle\psi_{f}|V_{i}|\psi_{i}\rangle,$ (3)

where $\psi_{i}$ is the unperturbed wavefunction of the initial channel which is given by,

$\displaystyle\psi_{i}=e^{i\vec{k_{i}}.\vec{r_{i}}}\phi_{i}(\vec{r}_{1T},\vec{r% }_{2T}),$ (4)

where $\vec{k}_{i}$ is the initial wave vactor of the incident projectile with respect to the target center of mass. Here $\phi_{i}(\vec{r}_{1T},\vec{r}_{2T})$ is the ground state of helium atoms. $\psi_{f}$ is the total scattering wavefunctioion which may be written as,

$\displaystyle\psi_{f}=e^{i\vec{k_{f}}.\vec{r_{f}}}\chi^{-}_{f}\phi_{f}(\vec{r}% _{1P},\vec{r}_{2P}),$ (5)

where $\chi^{-}_{f}$ is the distorted wave. $\vec{k}_{f}$ is the final wave vector. $\phi_{f}(\vec{r}_{1P},\vec{r}_{2P})$ is the final bound state of helium-like atom. The final bound state $\phi_{f}(\vec{r}_{1P},\vec{r}_{2P})$ on a set of two-electron hydrogenic configuration [14, 15] can be written as

$\displaystyle\phi_{f}(\vec{r}_{{1P}},\vec{r}_{{2P}})=\sum_{\lambda\leqslant% \lambda^{\prime}}a^{f}_{{\lambda,\lambda^{\prime}}}\phi^{I}_{{\lambda,\lambda^% {\prime}}}(\vec{r}_{{1P}},\vec{r}_{{2P}})=\sum_{\lambda<\lambda^{\prime}}\frac% {a^{f}_{{\lambda,\lambda^{\prime}}}}{\sqrt{2}}\sum_{m,m^{\prime}}\{(l_{\lambda% }m,l_{{\lambda^{\prime}}}m^{\prime}/LM)R_{\lambda}(\vec{r}_{{1P}})R_{{\lambda^% {\prime}}}(\vec{r}_{{2P}})Y^{m}_{{l_{\lambda}}}(\hat{r}_{{1P}})Y^{m^{\prime}}_% {{l_{{\lambda^{\prime}}}}}(\hat{r}_{{2P}})+(-1)^{S}(l_{{\lambda^{\prime}}}m^{% \prime},l_{\lambda}m/LM)R_{\lambda}(\vec{r}_{{2P}})R_{{\lambda^{\prime}}}(\vec% {r}_{{1P}})Y^{m}_{{l_{\lambda}}}(\hat{r}_{{2P}})Y^{m^{\prime}}_{{l_{{\lambda^{% \prime}}}}}(\hat{r}_{{1P}})\}+\frac{1+(-1)^{L+S}}{2}\sum_{\lambda}a^{f}_{{% \lambda,\lambda}}\sum_{m,m^{\prime}}(l_{\lambda}m,l_{{\lambda^{\prime}}}m^{% \prime}/LM)R_{\lambda}(\vec{r}_{{1P}})R_{{\lambda}}(\vec{r}_{{2P}})Y^{m}_{{l_{% \lambda}}}(\hat{r}_{{1P}})Y^{m^{\prime}}_{{l_{{\lambda^{\prime}}}}}(\hat{r}_{{% 2P}}),$

where $\lambda=(n_{\lambda},l_{\lambda})$ , $\lambda^{\prime}=(n_{{\lambda^{\prime}}},l_{{\lambda^{\prime}}})$ . Here $R_{\lambda}(\vec{r}_{{ip}}),Y^{m}_{{l_{\lambda}}}(\hat{r}_{{ip}})(i=1,2)$ are radial hydrogenic functions and spherical harmonics respectively. $(l_{\lambda}m,l_{{\lambda^{\prime}}}m^{\prime}/LM)$ is a Clebsch-Gordan coefficient. For example, the explicit form of singly and doubly excited states of two electron helium-like wavefunctions are given below:

(i) For $(1s2s)^{1}S$ :

$\displaystyle\phi^{f}_{{1s2s}}(\vec{r}_{{1p}},\vec{r}_{{2p}})=0.118\left(\frac% {z}{a_{0}}\right)^{3/2}\left(\frac{z}{2a_{0}}\right)^{3/2}\left[e^{-\frac{z}{a% _{0}}r_{{1p}}}\left(1-\frac{z}{2a_{0}}r_{{2p}}\right)e^{-\frac{z}{2a_{0}}r_{{2% p}}}+e^{-\frac{z}{a_{0}}r_{{2p}}}\left(1-\frac{z}{2a_{0}}r_{{1p}}\right)e^{-% \frac{z}{2a_{0}}r_{{1p}}}\right]$

(ii) For $(1s2p)^{1}P$ :

$\displaystyle\phi^{f}_{{1s2p}}(\vec{r}_{{1p}},\vec{r}_{{2p}})=0.193\left(\frac% {z}{a_{0}}\right)^{3/2}\left(\frac{z}{2a_{0}}\right)^{3/2}\left[e^{-\frac{z}{a% _{0}}r_{{1p}}}\left(\frac{z}{a_{0}}r_{{2p}}\right)e^{-\frac{z}{2a_{0}}r_{{2p}}% }\right.\left\{\left(\frac{3}{4\pi}\right)^{\frac{1}{2}}\cos\theta_{{2p}}+% \left(\frac{3}{8\pi}\right)^{\frac{1}{2}}e^{i\phi_{{2p}}}\sin\theta_{{2p}}-% \left(\frac{3}{8\pi}\right)^{\frac{1}{2}}e^{-i\phi_{{2p}}}\sin\theta_{{2p}}% \right\}+e^{-\frac{z}{a_{0}}r_{{2p}}}\left(\frac{z}{a_{0}}r_{{1p}}\right)e^{-% \frac{z}{2a_{0}}r_{{1p}}}\left.\left\{\left(\frac{3}{4\pi}\right)^{\frac{1}{2}% }\cos\theta_{{1p}}+\left(\frac{3}{8\pi}\right)^{\frac{1}{2}}e^{i\phi_{{1p}}}% \sin\theta_{{1p}}-\left(\frac{3}{8\pi}\right)^{\frac{1}{2}}e^{-i\phi_{{1p}}}% \sin\theta_{{1p}}\right\}\right]$

(iii) For $(2s2s)^{1}S$ :

$\displaystyle\phi^{f}_{{2s2s}}(\vec{r}_{{1p}},\vec{r}_{{2p}})=0.28\left(\frac{% z}{2a_{0}}\right)^{3}\left[\left(1-\frac{z}{2a_{0}}r_{{1p}}\right)\left(1-% \frac{z}{2a_{0}}r_{{2p}}\right)e^{-\frac{z}{2a_{0}}(r_{{1p}}+r_{{2p}})}\right]$

and

(iv) For $(2s2p)^{1}P$ :

$\displaystyle\phi^{f}_{{2s2p}}(\vec{r}_{{1p}},\vec{r}_{{2p}})=0.11(\frac{z}{2a% _{0}})^{3}\left[\left(\frac{z}{a_{0}}r_{{2p}}\right)\left(1-\frac{z}{2a_{0}}r_% {{1p}}\right)e^{-\frac{z}{2a_{0}}(r_{{1p}}+r_{{2p}})}\right.\left\{\left(\frac% {3}{4\pi}\right)^{\frac{1}{2}}\cos\theta_{{2p}}+\left(\frac{3}{8\pi}\right)^{% \frac{1}{2}}e^{i\phi_{{2p}}}\sin\theta_{{2p}}-\left(\frac{3}{8\pi}\right)^{% \frac{1}{2}}e^{-i\phi_{{2p}}}\sin\theta_{{2p}}\right\}+\left(\frac{z}{a_{0}}r_% {{1p}}\right)\left(1-\frac{z}{2a_{0}}r_{{2p}}\right)e^{-\frac{z}{2a_{0}}(r_{{1% p}}+r_{{2p}})}\left.\left\{\left(\frac{3}{4\pi}\right)^{\frac{1}{2}}\cos\theta% _{{1p}}+\left(\frac{3}{8\pi}\right)^{\frac{1}{2}}e^{i\phi_{{1p}}}\sin\theta_{{% 1p}}-\left(\frac{3}{8\pi}\right)^{\frac{1}{2}}e^{-i\phi_{{1p}}}\sin\theta_{{1p% }}\right\}\right].$

The two different distorted waves considered for exit channel are the four-body model of (i) MTCDW-4B [8, 32, 33] and (ii) BCCIS-4B [16, 17, 18] approximation which we shall refer as complete BCCIS-4B approximation. In addition, we have also calculated the cross sections with modification of complete BCCIS-4B [16, 17, 18] using the asymptotic Coulomb logarithmic phase instead of full Coulomb interaction for the relative motion of two colliding nuclei. Hereafter we shall refer it as approximate BCCIS-4B method. Here, the MTCDW-4B method is the extension of TCDW-4B method originally proposed by Crothers and Dunseath [8] where the perturbation $V_{i}=V_{{P1}}+V_{{P2}}$ from the TCDW-4B method is presently replaced by the modified perturbation potential $V_{i}=V_{{PT}}+V_{{P1}}+V_{{P2}}$ in the MTCDW-4B method. Here $V_{{P1}}$ , $V_{{P2}}$ and $V_{{PT}}$ represents the projectile-active electron(s) and inter-nuclear interaction potential. Although the present MTCDW-4B method does not satisfy either the CBC in the entrance and exit channels. It is noted that the inter-nuclear potential $(V_{{PT}})$ in $V_{i}$ is not counter-balanced by the missing function ${{{1}}}F_{{1}}\{-i\alpha_{{3}};1;i(\vec{k_{{f}}}.\vec{R}+k_{{f}}R)\}$ which is replaced by unity in the present MTCDW-4B method. However the explicit form of $\chi^{-}_{f}$ becomes

$\displaystyle\chi^{(\textit{MTCDW-4B})-}_{{f}}=e^{{\frac{\pi}{2}(\alpha_{{1}}+% \alpha_{{2}})}}\Gamma(1+i\alpha_{{1}})\Gamma(1+i\alpha_{{2}}){}_{{{1}}}F_{{1}}% \{-i\alpha_{{1}};1;-i(\vec{v_{{f}}}.\vec{r}_{{1T}}+v_{{f}}r_{{1T}})\}{{{1}}}F_% {{1}}\{-i\alpha_{{2}};1;-i(\vec{v_{{f}}}.\vec{r}_{{2T}}+v_{{f}}r_{{2T}})\}$ (6)

and

$\displaystyle\chi^{(\textit{BCCIS-4B})-}_{{f}}=e^{\frac{\pi}{2}(\alpha_{{1}}+% \alpha_{{2}}-\alpha_{{3}})}\Gamma(1+i\alpha_{{1}})\Gamma(1+i\alpha_{{2}})% \Gamma(1-i\alpha_{{3}}){{{1}}}F_{{1}}\{-i\alpha_{{1}};1;-i(\vec{v_{{f}}}\vec{r% }_{{1T}}+v_{{f}}r_{{1T}})\}{{{1}}}F_{{1}}\{-i\alpha_{{2}};1;-i(\vec{v_{{f}}}.% \vec{r}_{{2T}}+v_{{f}}r_{{2T}})\}{{{1}}}F_{{1}}\{i\alpha_{{3}};1;-i(\vec{k_{{f% }}}.\vec{r}_{{i}}+k_{{f}}r_{{i}})\},$ (7)

where $\alpha_{{1}}=\alpha_{{2}}=\frac{z_{{T}}}{v_{{f}}},\alpha_{{3}}=\frac{z_{{P}}z_% {{T}}}{bv_{{f}}}$ with $v_{f}$ being the final relative velocity. Here we have used on-shell energy conservation relations such as,

$\displaystyle E_{{i}}=E_{{f}}=\frac{k^{2}_{{i}}}{2\mu_{{i}}}+\epsilon_{{i}}=% \frac{k^{2}_{{f}}}{2\mu_{{f}}}+\epsilon_{{f}},$ (8)

where $\epsilon_{{i}}(\epsilon_{{f}})$ is the binding energy of the initial (final) bound state of helium-like atom. Finally, the transition amplitude in the MTCDW-4B, the complete BCCIS-4B and the approximate BCCIS-4B methods may be written as,

$\displaystyle T^{(\textit{MTCDW-4B})-}_{\textit{if}}=N_{{1}}\int\!\!\int\!\!% \int\!\!{d\vec{r}_{{1T}}}{d\vec{r}_{{2T}}}{d\vec{R}}e^{i\vec{k}_{{i}}.\vec{r}_% {{i}}-i\vec{k}_{{f}}.\vec{r}_{{f}}}{}_{{1}}F_{{1}}\{i\alpha_{{1}};1;i(\vec{v}_% {{f}}.\vec{r}_{{1T}}+v_{{f}}r_{{1T}})\}{}_{{1}}F_{{1}}\{i\alpha_{{2}};1;i(\vec% {v}_{{f}}.\vec{r}_{{2T}}+v_{{f}}r_{{2T}})\}\phi^{*}_{{f}}(\vec{r}_{{1P}},\vec{% r}_{{2P}})\phi_{{i}}(\vec{r}_{{1T}},\vec{r}_{{2T}})\left(\frac{z_{{P}}z_{{T}}}% {R}-\frac{z_{{P}}}{r_{{1P}}}-\frac{z_{{P}}}{r_{{2P}}}\right),$ (9)

where $N_{{1}}=e^{{\frac{\pi}{2}(\alpha_{{1}}+\alpha_{{2}})}}\Gamma(1-i\alpha_{{1}})% \Gamma(1-i\alpha_{{2}})$ .

$\displaystyle T^{(\textit{BCCIS-4B})-}_{\textit{if}}=N_{{2}}\int\!\!\int\!\!% \int\!{d\vec{r}_{{1T}}}{d\vec{r}}_{{2T}}{d\vec{R}}e^{i\vec{k}_{{i}}.\vec{r}_{{% i}}-i\vec{k}_{{f}}.\vec{r}_{{f}}}{}_{1}F_{{1}}\{i\alpha_{{1}};1;i(\vec{v}_{{f}% }.\vec{r}_{{1T}}+v_{{f}}r_{{1T}})\}{}_{{{1}}}F_{{1}}\{i\alpha_{{2}};1;i(\vec{v% }_{{f}}.\vec{r}_{{2T}}+v_{{f}}r_{{2T}})\}{}_{{{1}}}F_{{1}}\{-i\alpha_{{3}};1;i% (\vec{k_{{f}}}.\vec{R}+k_{{f}}R)\}\phi_{{f}}^{*}(\vec{r}_{{1P}},\vec{r}_{{2P}}% )\phi_{{i}}(\vec{r}_{{1T}},\vec{r}_{{2T}})\left(\frac{z_{{P}}z_{{T}}}{R}-\frac% {z_{{P}}}{r_{{1P}}}-\frac{z_{{P}}}{r_{{2P}}}\right),$ (10a)

where $N_{{2}}=e^{\frac{\pi}{2}(\alpha_{{1}}+\alpha_{{2}}-\alpha_{{3}})}\Gamma(1-i% \alpha_{{1}})\Gamma(1-i\alpha_{{2}})\Gamma(1+i\alpha_{{3}})$ and

$\displaystyle T^{(\textit{BCCIS-4B-approx})-}_{\textit{if}}=N_{{1}}\int\!\!% \int\!\!\int\!{d\vec{r}_{{1T}}}{d\vec{r}}_{{2T}}{d\vec{R}}e^{i\vec{k}_{{i}}.% \vec{r}_{{i}}-i\vec{k}_{{f}}.\vec{r}_{{f}}}{{{1}}}F_{{1}}\{i\alpha_{{1}};1;i(% \vec{v}_{{f}}.\vec{r}_{{1T}}+v_{{f}}r_{{1T}})\}{}_{{{1}}}F_{{1}}\{i\alpha_{{2}% };1;i(\vec{v}_{{f}}.\vec{r}_{{2T}}+v_{{f}}r_{{2T}})\}(\vec{k_{{f}}}.\vec{R}+k_% {{f}}R)^{i\alpha_{3}}\phi_{{f}}^{*}(\vec{r}_{{1P}},\vec{r}_{{2P}})\phi_{{i}}(% \vec{r}_{{1T}},\vec{r}_{{2T}})\left(\frac{z_{{P}}z_{{T}}}{R}-\frac{z_{{P}}}{r_% {{1P}}}-\frac{z_{{P}}}{r_{{2P}}}\right),$ (10b)

where $r_{i}\approx R$ and $r_{f}\approx R$ are taken in an approximation of the order of $1/M_{T}$ and $1/M_{P}$ respectively.

The integral representation of confluent hypergeometric function and logarithmic Coulomb phase factor [12] are given by,

$\displaystyle{}_{{1}}F_{{1}}(\alpha;1;z)=\frac{1}{2\pi i}\oint_{{\Gamma}}(t-1)% ^{-i\alpha}t^{i\alpha-1}e^{zt}{dt}.$ (11)

and

$\displaystyle(\vec{k_{{f}}}.\vec{R}+k_{{f}}R)^{i\alpha_{3}}=\frac{e^{\frac{\pi% }{2}\alpha_{3}}\Gamma(1+i\alpha_{{3}})}{2\pi i}\oint_{\Gamma}dt_{3}(-t_{3})^{-% i\alpha_{3}-1}e^{-i(\vec{k_{{f}}}.\vec{R}+k_{{f}}R)t_{3}}.$ (12)

Using Eqs (11) and (12), the Fourier transform identity as

$\displaystyle e^{-\lambda r}=\frac{\lambda}{\pi^{2}}\int\frac{e^{i\vec{p}.\vec% {r}}}{(p^{2}+\lambda^{2})^{2}}{d\vec{p}},$ (13)

the Feynman parametrization integral

$\displaystyle\frac{1}{A^{n}B^{m}}=\frac{(n+m-1)!}{(n-1)!(m-1)!}\int^{1}_{{0}}{% dx}x^{n-1}(1-x)^{m-1}[Ax+(1-x)B]^{-n-m},$ (14)

where $n, m$ are certain integer numbers, and following Lewis integral [34], the Eqs (9), (10a) and (10b) becomes

$\displaystyle T^{(\textit{MTCDW-4B})-}_{\textit{if}}=32\pi^{3}N_{{1}}C\left(% \frac{e^{-\pi\alpha_{{1}}}-e^{\pi\alpha_{{1}}}}{2\pi i}\right)\int^{1}_{{0}}{{% dt}_{{1}}}t_{{1}}^{i\alpha_{{1}}}(1-t_{{1}})^{-i\alpha_{{1}}}\times D(\delta_{% {1}},\delta_{{2}},\lambda_{{1}},\lambda_{{2}},\epsilon_{{i}})\int^{1}_{{0}}% \frac{{dx}}{\Delta}\int^{\infty}_{{0}}\sigma^{i\alpha_{{2}}-1}_{{0}}(\sigma_{{% 0}}+\sigma_{{1}})^{-i\alpha_{{2}}}{dy},$ (15)

$\displaystyle T^{(\textit{BCCIS-4B})-}_{\textit{if}}=32\pi^{3}N_{{2}}C\left(% \frac{e^{-\pi\alpha_{{1}}}-e^{\pi\alpha_{{1}}}}{2\pi i}\right)\int^{1}_{{0}}{{% dt}_{{1}}}t_{{1}}^{i\alpha_{{1}}}(1-t_{{1}})^{-i\alpha_{{1}}}D(\delta_{{1}},% \delta_{{2}},\lambda_{{1}},\lambda_{{2}},\epsilon_{{i}})\int^{1}_{{0}}\frac{{% dx}}{\Delta}\int^{\infty}_{{0}}\sigma^{i\alpha_{{2}}-i\alpha_{{3}}-1}_{{0}}(% \sigma_{{0}}+\sigma_{{1}})^{-i\alpha_{{2}}}(\sigma_{{0}}+\sigma_{{2}})^{i% \alpha_{{3}}}{}_{{2}}F_{{1}}\left\{i\alpha_{{2}};-i\alpha_{{3}};1;\frac{\sigma% _{1}\sigma_{2}-\sigma_{0}\sigma_{3}}{(\sigma_{0}+\sigma_{1})(\sigma_{0}+\sigma% _{2})}\right\}{dy},$ (16a)

and

$\displaystyle T^{(\textit{BCCIS-4B-approx})-}_{\textit{if}}=32\pi^{3}N_{{2}}C% \left(\frac{e^{-\pi\alpha_{{1}}}-e^{\pi\alpha_{{1}}}}{2\pi i}\right)\int^{1}_{% {0}}{{dt}_{{1}}}t_{{1}}^{i\alpha_{{1}}}(1-t_{{1}})^{-i\alpha_{{1}}}D(\delta_{{% 1}},\delta_{{2}},\lambda_{{1}},\lambda_{{2}},\epsilon_{{i}})\int^{1}_{{0}}% \frac{{dx}}{\Delta}\int^{\infty}_{{0}}\sigma^{-i\alpha_{{3}}-1}_{{0}}\sigma^{i% \alpha_{3}+i\alpha_{2}}_{2}(\sigma_{{2}}+\sigma_{{3}})^{-i\alpha_{{2}}}{}_{{2}% }F_{{1}}\left\{i\alpha_{{2}};1+i\alpha_{{3}};1;\frac{\sigma_{0}\sigma_{3}-% \sigma_{1}\sigma_{2}}{\sigma_{0}(\sigma_{2}+\sigma_{3})}\right\}{dy},$ (16b)

where $D(\delta_{1},\delta_{2},\lambda_{1},\lambda_{2},\epsilon_{i})$ is a parametric differential operator. The form of the operator is given by

$\displaystyle D(\delta_{1},\delta_{2},\lambda_{1},\lambda_{2},\epsilon_{i})=% \frac{\partial^{4}}{\partial\delta_{1}\partial\delta_{2}\partial\lambda_{1}% \partial\lambda_{2}}-\frac{\partial^{4}}{\partial\delta_{1}\partial\delta_{2}% \partial\lambda_{2}\partial\epsilon_{i}}-\frac{\partial^{4}}{\partial\delta_{1% }\partial\delta_{2}\partial\lambda_{1}\partial\epsilon_{i}}.$ (17)

Here the constant C in Eqs (15), (16a) and (16b) originates from the initial and final bound state wavefunctions. The values of $\sigma_{0}$ , $\sigma_{1}$ , $\sigma_{2}$ and $\sigma_{3}$ are given in details in the Appendix A for the complete BCCIS-4B and the Appendix B for the approximate BCCIS-4B. In addition, the explicit results of the applications of the partial differential operator $D(\delta_{1},\delta_{2},\lambda_{1},\lambda_{2},\epsilon_{i})$ are provided in the appendix to complete the analytical calculations. The Lewis integral with infinite upper limit and the Feynman integral from 0 to 1 have been calculated numerically by the 36-point Gauss-Legendre quadrature method with suitable transformation to achieve an accuracy of 0.001%. Range of $t_{1}$ -integral from 0 to 1 has been divided into eleven unequal intervals with shorter parts on either side of the two limits and each interval has been evaluated by 34-point Gauss-Laguerre quadrature method to achieve an accuracy of 0.01%. However, it may be mentioned that each interval has varying quadrature points for convergence but we have used the highest quadrature points for all intervals. The detailed evaluation of these integrals are found elsewhere [12, 16, 17, 18, 32]. In the centre of mass frame, the DCS for DC is given by,

$\displaystyle\sigma(\theta,E)=\frac{{d\sigma}}{{d\Omega}}=\frac{\mu_{{i}}\mu_{% {f}}}{4\pi^{2}}\frac{k_{{f}}}{k_{{i}}}|T_{\textit{if}}|^{2},$ (18)

where $\theta$ is the projectile scattering angle and the TCS is,

$\displaystyle\sigma_{{Total}}(E)=2\pi\int^{+1}_{{-1}}\sigma(\theta,E){d(\cos% \theta)}.$ (19)

Finally, the TCS is obtained by numerical integration over the scattering angles using 48-point Gauss-Legendre quadrature method with an accuracy of 0.1%.

Figure 1.

(a) Variation of TCS for DC into the ground state with energies in collision of fast $\alpha$ -particles with helium atoms in their ground state. Theory: solid curve, complete BCCIS-4B results (using Eq. (16a)) and dashed curve, approximate BCCIS-4B results (using Eq. (16b)) obtained by non-correlated one-parameter Hylleraas wavefunction [36]; dotted curve, BCIS-4B results of Belkic [11, 12]. (b) Variation of TCS for DC into ground state with energies in collision of fast $\alpha$ -particles with helium atoms in their ground state. Theory: solid curve, 1: complete BCCIS-4B results (using Eq. (16a)); short dash-dotted curve, 2: present MTCDW-4B results (using Eq. (15)); dotted curve, 3: BCIS-4B results using non-correlated one-parameter orbital for the helium bound state [11, 12]; dash-dotted curve, 4: BDW-4B results [27]; short-dotted curve, 5: CB1-4B results [41]; dash-dot-dotted curve, 6: CDW-4B results [6]; short dashed curve, 7: approximate BCCIS-4B (using Eq. (16b)). Experiment: solid circle, experimental results of DuBois [38]; solid square, results of Schuch et al. [40]; solid up triangle, results of de Castro et al. [39]; solid down triangle, results of Berkner et al. [37]. (c) Variation of TCS for DC into both ground and excited states with energies in collision of fast $\alpha$ -particles with helium atoms in their ground state. Theory: solid curve, 1: complete BCCIS-4B results (using Eq. (16a)); dashed curve, 2: present MTCDW-4B results (using Eq. (15)); dash dotted curve, 3: TC-BGM results [24]; dotted curve, 4: BCCIS-4B results using non-correlated one-parameter orbital for the helium atom in their ground state [17]; dash-dot-dotted curve, 5: CDW-4B results [14]. Experiment: solid circle, experimental results of DuBois [38]; solid square, results of Schuch et al. [40]; solid up triangle, results of de Castro et al. [39]; solid down triangle, results of Berkner et al. [37].

Figure 2.

Variation of TCS (in $cm^{2}$ ) for DC into both ground and excited states as a function of the incident energy E (keV) for $\textit{Li}^{3+}-\textit{He}$ collision. Theory: solid curve,1: complete BCCIS-4B results (using Eq. (16a)); dashed curve, 2: MTCDW-4B results (using Eq. (15)); dotted curve, 3: CB1-4B results [42]; dash dotted curve, 4: CDW-4B results [14]; dash-dot-dotted curve, 5: BCCIS-4B results using non-corrected one-parameter orbital for the helium bound state [17] (only ground state capture). Experiment: solid circle, experimental results [43].

Figure 3.

Relative contribution of ground state ( $(1s1s)^{1}S$ ) and excited states (singly and doubly) in total DC cross section as a function of incident projectile energy for $\textit{He}^{2+}-\textit{He}$ collision. Theory: solid curve, 1: complete BCCIS-4B results (for ground state); dash-dotted curve, 2: complete BCCIS-4B results (excited states); dashed curve, 3: CDW-4B results [14] (ground state); short-dotted curve, 4: CDW-4B results [14] (excited states).

Figure 4.

Same as Fig. 3, but for $\textit{Li}^{3+}$ impact on He.

3. Results and discussion

In this section, theoretical results for DC by the impact of fast bare ions ( $\textit{He}^{2+}$ and $\textit{Li}^{3+}$ ) on helium atoms in ground state have been calculated by using MTCDW-4B and complete BCCIS-4B approximation. In addition, we have also computed the cross sections with modification of BCCIS-4B using the asymptotic Coulomb logarithmic phase instead of full Coulomb interaction. Present results are compared with the previous theoretical results obtained by different models and with experimental data. The variation of TCS for DC with the incident energy of the projectile ions ( $\textit{He}^{2+}$ and $\textit{Li}^{3+}$ ) are reported in graphical form in Figs 1a–c and 2, respectively. Relative contribution of ground state and excited states in total DC cross sections with projectile energy are displayed in Figs 3 and 4 respectively. In addition, DCS for DC are also studied in both methods (MTCDW-4B and complete BCCIS-4B) for $\textit{He}^{2+}-\textit{He}$ collision at different projectile energies which are shown in Figs 5a and b and 6–8 respectively. The present results are obtained using 4-parameter radially-correlated wave function [35] as

$\displaystyle\phi_{{i}}(\vec{r}_{{1T}},\vec{r}_{{2T}})=\frac{N}{4\pi}(Ae^{-% \alpha{r}_{{1T}}}+Be^{-\beta{r}_{{1T}}})(Ae^{-\alpha{r}_{{2T}}}+Be^{-\beta{r}_% {{2T}}}),$

where the normalization constant N is given by $N=4/[{A^{2}}/{\alpha^{3}}+{B^{2}}/{\beta^{3}}+{16AB}/{(\alpha+\beta)^{3}}]$ . Here A $=$ 2.60505, B $=$ 2.08144, $\alpha=$ 1.41, and $\beta=$ 2.61, consequently N $=$ 1. However, for Figs 1a and 5a, we have used the one-parameter helium wavefunction of Hylleraas [36] for the initial and final bound state which is given by

$\displaystyle\phi_{{i,f}}(\vec{r}_{1},\vec{r}_{2})=\left(\frac{\delta^{3}}{\pi% }\right)e^{-\delta(r_{1}+r_{2})}$

with effective charge $\delta=1.6875$ and the corresponding binding energy $\epsilon_{{i,f}}=2.9037244$ a.u.

Table 1
CS (in $cm^{2}$ ) for ground-to-ground state DC (using the one-parameter helium wavefunction of Hylleraas [36] in the initial and final state) as a function of the incident energy (keV) for the collision $\textit{He}^{2+}-\textit{He}$

Energy (keV)	BCCIS-4B model using Eq. (16a)	BCCIS-4B model using Eq. (16b)
100	5.84( $-$ 17)	2.83( $-$ 16)
200	3.22( $-$ 17)	9.42( $-$ 17)
240	2.55( $-$ 17)	6.06( $-$ 17)
320	1.68( $-$ 17)	2.82( $-$ 17)
400	1.15( $-$ 17)	1.43( $-$ 17)
600	4.00( $-$ 18)	2.82( $-$ 18)
800	2.05( $-$ 18)	6.69( $-$ 19)
1000	4.37( $-$ 19)	1.74( $-$ 19)
1600	1.11( $-$ 20)	6.50( $-$ 21)
2000	1.01( $-$ 21)	9.09( $-$ 22)
3200	1.85( $-$ 23)	2.42( $-$ 23)
4000	2.21( $-$ 24)	2.72( $-$ 24)
8000	3.62( $-$ 27)	5.65( $-$ 27)
10000	4.21( $-$ 28)	5.90( $-$ 28)

Figure 5.

(a) Differential DC cross sections in ground state for $\textit{He}^{2+}-\textit{He}$ collisions at projectile energy 150 keV/amu. Theory: solid curve, complete BCCIS-4B results (using Eq. (16a)) and dashed curve, approximate BCCIS-4B (using Eq. (16b)) obtained by non-correlated one-parameter Hylleraas wavefunction [36]. (b) Differential DC cross sections in ground state for $\textit{He}^{2+}-\textit{He}$ collisions at projectile energy 150 keV/amu. Theory: solid curve, 1: complete BCCIS-4B results (using Eq. (16a)); dashed curve, 2: MTCDW-4B results (using Eq. (15)); short dotted curve, 3: BDW-4B results [25]; dash dotted curve, 4: approximate BCCIS-4B results (using Eq. (16b)). Experiment: solid circle, experimental results of Schoffler et al. [25].

Figure 6.

Same of Fig. 5, but for 375 keV/amu $\textit{He}^{2+}$ impact on He. Theory: solid curve, 1: complete BCCIS-4B results (using Eq. (16a)); dashed curve, 2: MTCDW-4B results (using Eq. (15)); dotted curve, 3: CB1-4B results [27]; dash-dot-dotted curve, 4: BCIS-4B results [12]; dash dotted curve, 5: BDW-4B results [27]; short dotted curve; 6: approximate BCCIS-4B results (using Eq. (16b)). Experiment: solid circle, experimental results of Schuch et al. [40] (incuding double capture into all bound states of He).

3.1 Total cross sections (TCS)

Projectile energy variation of DC cross sections into ground state for $\textit{He}^{2+}-\textit{He}$ collision have been displayed in Fig. 1a and the numerical cross sections are given in Table 1. Here we have plotted only two results: one for complete BCCIS-4B (using Eq. (16a)) and other for approximate BCCIS-4B (using Eq. (16b)). We have used non-correlated one-parameter wavefunction [36] of He in both channels. We find that the cross sections from the approximate BCCIS-4B method (dashed curve) are noticeably larger than those from the complete BCCIS-4B (solid curve) at 100–500 keV with the opposite pattern at 500–2000 keV. However, the discrepancy in the two set of results appears to be much less pronounced at higher energies (above 2000 keV). It can be seen from Fig. 1a, that the BCIS-4B results underestimate the BCCIS-4B (complete and approximate) in the energy range from 100 to 2000 keV and slightly overestimate the present results at higher energies. Though the transition amplitude of the BCIS-4B and the present approximate BCCIS-4B (Eq. (10b)) contains asymptotic logarithmic coulomb phase wavefunction, but we find that the two sets of results are different at lower energy region. This may be due to the following reasons: (i) the plane wave part of Eq. (10b) i.e., $e^{i(\vec{k_{i}}.\vec{r_{i}}-\vec{k_{f}}.\vec{r_{f}})}$ is replaced by $e^{i(\vec{k_{i}}-\vec{k_{f}}).\vec{R}}$ with the eikonal approximation $\vec{r_{i}}\approx{\vec{r_{f}}}\approx{\vec{R}}$ $(M_{{P,T}}\gg 1)$ and (ii) after simplification the final form of the transition amplitude in the approximate BCCIS-4B (Eq. (16b)) and the BCIS-4B [11, 12] are not identical. Figure 1b displays the variation of the TCS for DC into ground state for the process of symmetric collision as a function of incident projectile energy ranging from 100 keV to 10 MeV. Our data obtained from complete BCCIS-4B (solid curve), approximate BCCIS-4B (short dash-dotted curve) and MTCDW-4B (dashed curve) are compared with the measurements of Berkner et al. [37], DuBois [38], de Castro et al. [39], Schuch et al. [40] and the BDW-4B [27], CDW-4B results of Belkic [6], boundary corrected four-body first Born (CB1-4B) [41] and BCIS-4B results of Belkic [12]. The BCIS-4B model takes account of the continuum states of both electrons with either the target ion in prior form and the projectile ion in post form together with the relative motion of the colliding nuclei being described by logarithmic Coulomb phase in either form of the transition amplitude. The present results in the prior form of the complete BCCIS-4B have good agreement with the experimental results in the entire range of energy, whereas the MTCDW-4B results have a trend of departure in the energy range between 300 keV to 3000 keV. This is due to the inclusion of projectile target nucleus Coulomb distortion in BCCIS-4B approximation. In both BCCIS-4B and MTCDW-4B model, distortion in the final channel related to Coulomb continuum states of the active electrons in the field of residual target ions are included. The results obtained by Eq. (16b) using asymptotic Coulomb logarithmic phase (dash-short curve:7) is closer to the experimental results of Dubois [38] (solid circle) in the whole energy range except at 700 keV. It may also be seen from Fig. 1b that the approximate BCCIS-4B results (short-dashed curve:7) have almost similar energy variation of cross sections within 35% to those of the complete BCCIS-4B method (solid curve:1) at impact energies ranging from 400 to 2000 keV. At lower incident energies the approximate BCCIS-4B results are larger than the complete BCCIS-4B and the measurement of Berkner et al. [37] (solid down triangle). It is also noted that the approximate BCCIS-4B (short-dashed curve:7) have trend of departure from both the complete BCCIS-4B, BDW-4B (dash-dotted curve:4) [27] and BCIS-4B (dotted curve:3) [11, 12] above 2000 keV as collision energy increases. In both these methods, we have used the non-correlated one-parameter wavefunction [36] for the ground state of helium atom in both channels. We also note that the CB1-4B results of Belkic overestimate, whereas the CDW-4B results [6] underestimate the experimental data. This is expected because the CB1-4B model does not include the effect of intermediate continuum states. The results obtained from the BCIS-4B model developed by Belkic [12] from 900 keV to 3000 keV have nice agreement with the experimental findings of de Castro et al. [39]. Hence, by comparing different theoretical models with CB1-4B model, we would learn that the relative importance of intermediate ionization electronic continua as the impact energy increases. However the second order BDW results show good agreement with the experimental finding of de Castro et al. [39]. The BDW model is a hybrid-type model which in the entrance and exit channel coincides, respectively, with the CDW-4B and CB1-4B methods. All methods (CB1, CDW, BCCIS, BCIS, BDW) satisfy the correct boundary conditions in the entrance and exit channels respectively. However, unlike the CB1-4B method, four-body version of the BDW, BCCIS, BCIS, CDW, MTCDW methods take full account of the continuum intermediate states of both electrons in either of the entrance and exit channel. In Fig. 1c, we have presented the calculated TCS for DC into both ground and excited states as a function of incident projectile energy from 100 to 10,000 keV. Our data are compared with the measurements [37, 38, 39, 40] and the theoretical results of prior from of BCCIS-4B [17] results (dotted curve: 4) using non-correlated one-parameter orbital for the He atom in their ground state, CDW-4B results (dash-dot-dotted curve: 5) of Gayet et al. [14], TC-BGM results (dash-dotted curve: 3) of Zapukhlyak and Kirchner [24]. In both these methods (complete BCCIS-4B and

Table 2
TCS (in $cm^{2}$ ) for DC into both ground and excited states as a function of the incident energy (keV) for the collision $\textit{He}^{2+}-\textit{He}$

Energy (keV)	BCCIS-4B (one-parameter)	BCCIS-4B (4-parameter)	TCDW-4B (4-parameter)
200	4.00( $-$ 17)	4.85( $-$ 17)	4.33( $-$ 17)
300	2.25( $-$ 17)	2.00( $-$ 17)	2.54( $-$ 17)
400	1.20( $-$ 17)	9.03( $-$ 18)	1.50( $-$ 17)
500	8.55( $-$ 18)	7.85( $-$ 18)	8.19( $-$ 18)
1000	8.44( $-$ 19)	1.36( $-$ 19)	3.77( $-$ 19)
2000	1.12( $-$ 21)	1.11( $-$ 21)	2.26( $-$ 21)
3000	3.21( $-$ 23)	4.36( $-$ 23)	7.82( $-$ 21)
4000	4.49( $-$ 24)	6.48( $-$ 24)	8.12( $-$ 24)

MTCDW-4B), we have used the 4-parameter radially correlated Byron-Joachain [35] wavefunction for ground state of helium atom. It may also be seen from Fig. 1c that the results obtained by TC-BGM method of Zapukhlyak and Kirchner [24] overestimate the experimental data in the whole range of projectile energies. We have also compared our results obtained from one-parameter Hylleraas wavefunction [36] (dotted curve) with corresponding values obtained by using the 4-parameter Byron-Joachain wavefunction [35] (solid curve). As is seen from Fig. 1c, both sets of the results are closer to the experiments at higher impact energies, whereas at intermediate energies ranging from 400 keV–2000 keV the results obtained from BCCIS-4B using one-parameter uncorrelated wavefunction for ground state of He overestimate the present BCCIS-4B using 4-parameter wavefunction and the experimental results. The results for TCS obtained by different methods using one and four-parameter wavefunction of He are given in Table 2. In Fig. 2, we compare our BCCIS-4B results for $\textit{Li}^{3+}-\textit{He}$ collision with other theoretical [14, 18, 42], and experimental results [43]. The present BCCIS-4B and MTCDW-4B results are seen to be in good agreement with the measurement of Shah and Gilbody [43]. The CB1-4B results in post form of Bellkic [42] with a full account of the long range Coulomb effects arising from the relative motion of the scattering aggregates have satisfactory agreement with the measurement [43] at incident energy ranging from 250 to 2000 keV, although the calculation have been done only for ground state capture. We clearly observe that the theoretical data obtained by CDW-4B model tend to underestimate the present results and the experimental data of Shah and Gilbody [43]. In CDW-4B model, the TCS has been obtained by summing over the contributions from ground state and excited states. We have shown the relative contribution of ground state ( $(1s1s)^{1}S$ ), sum of singly excited states ( $(1s2s)^{1}S,(1s2p)^{1}P,(1s3s)^{1}S,(1s3p)^{1}P,(1s3d)^{1}D$ ) and doubly excited states ( $(2s2s)^{1}S,(2s2p)^{1}P$ ) in total DC cross sections for $\textit{He}^{2+}-He$ collision in Fig. 3 and for $\textit{Li}^{3+}-\textit{He}$ collision in Fig. 4 respectively. From Fig. 3, we find that the maximum contribution of DC cross sections comes from ground state over the entire energy range because the ground state is a resonating state. It is observed that, with increasing projectile energy, contributions from excited states gradually diminishes with a maximum of 50% contribution comes from excited states at 100 keV. This is expected because at higher energy, capture takes place predominantly close to the nucleus, i.e. in a region where high electron momentum transfer takes place. This feature also supports the results obtained by CDW-4B approximation. For $\textit{Li}^{3+}-\textit{He}$ collision process, we have also plotted the relative contributions from different states as a function of projectile energies in Fig. 4. Here we find that, at smaller impact energies (say 400 keV, close to the range of intermediate energies), the significant contribution to the TCS comes from excited states which contribute approximately to 35% of the TCS. However, difference between cross sections for the ground state and for excited states diminishes with increasing projectile energy, which is verified by other theoretical results obtained by CDW-4B model [14].

3.2 Differential cross sections (DCS)

Next we turn our attention to DCS which provide a more sensitive test for theoretical models. The results from the BCCIS-4B and MTCDW-4B method for DCS in ground state of helium atom at 150 keV/amu and 375 keV/amu are shown in Figs 5a–b and 6 respectively. In Fig. 5a, we have presented the calculated DCS for DC in ground state of He by 150 keV/amu $\alpha$ -particle impact on He both in complete BCCIS-4B (using Eq. (16a)) and approximate BCCIS-4B (using Eq. (16b)) methods as a function of projectile scattering angle from 0 to 2 mrad. It is evident from this figure that the approximate BCCIS-4B always underestimate the complete BCCIS-4B results in the whole range of projectile scattering angle. Around the scattering angle $\theta_{{lab}}\approx$ 0.5 mrad, a slight shoulder is seen in the dashed curve of approximate BCCIS-4B model, since the impact energy is too small to exhibit a net effect of the Thomas double scattering [21]. The DCS for DC in prior form of both complete BCCIS-4B and approximate BCCIS-4B methods are shown in Fig. 5b at incident energy of 150 keV/amu and compared with the BDW-4B results [25] and experimental results of Schoffler et al. [25]. The results for DCS obtained by the complete BCCIS-4B are given in Table 3 at 150 keV/amu. The BCCIS-4B results using correlated multi-parameters of Byron and Joachain orbitals [35] show excellent agreement with the experimental data [25] in the whole range of projectile scattering angles at projectile energy of 150 keV/amu.

Table 3
DCS ( $cm^{2}/sr$ ) into the ground state as a function of the scattering angle $\theta$ (mrad) at the incident energy 150 keV/amu for the DC cross sections in $\textit{He}^{2+}-\textit{He}(1s^{2})$ collisions

Scattering angle $\theta$ (mrad)	DCS (cm ${}^{2}$ /sr)	Scattering angle $\theta$ (mrad)	DCS (cm ${}^{2}$ /sr)
0	2.530E-11	0.3303	8.050E-13
0.0039	2.510E-11	0.3320	7.767E-13
0.0091	2.460E-11	0.3500	6.326E-13
0.0144	2.420E-11	0.3650	5.503E-13
0.0249	2.300E-11	0.3790	4.794E-13
0.0301	2.240E-11	0.4042	4.000E-13
0.0407	2.100E-11	0.4340	2.959E-13
0.0513	1.800E-11	0.4650	2.520E-13
0.0621	1.650E-11	0.4850	2.246E-13
0.0731	1.580E-11	0.5000	2.120E-13
0.0842	1.370E-11	0.5500	1.826E-13
0.0955	1.200E-11	0.6066	1.600E-13
0.1070	1.100E-11	0.7150	1.052E-13
0.1180	9.000E-12	0.7540	9.000E-14
0.1200	8.376E-12	0.8000	7.202E-14
0.1370	7.050E-12	0.8900	4.930E-14
0.1500	5.934E-12	1.0000	3.260E-14
0.1630	5.320E-12	1.2000	1.753E-14
0.1790	3.924E-12	1.3000	1.242E-14
0.1860	3.419E-12	1.4000	8.799E-15
0.2048	2.780E-12	1.5000	6.700E-15
0.2150	2.260E-12	1.6000	5.621E-15
0.2300	1.969E-12	1.7000	4.267E-15
0.2527	1.510E-12	1.8000	3.717E-15
0.2660	1.290E-12	2.0000	2.400E-15
0.2891	1.010E-12

Figure 7.

DCS for DC as a function of scattering angle for projectile energy 60 keV/amu, $\textit{He}^{2+}-\textit{He}$ collision. Theory: solid curve, complete BCCIS-4B results (using Eq. (16a)); dotted curve, TC-BGM results [24]; dashed curve, MTCDW-4B results (using Eq. (15)). Experiment: solid circle, experimental results of Schoffler et al. [25, 44].

Figure 8.

Same of Fig. 7, but for 300 keV/amu $\textit{He}^{2+}$ impact of on He.

As can be seen from Fig. 5b, at large scattering angles (nuclear scattering) all theoretical models except MTCDW-4B (dashed curve) and BCCIS-4B with asymptotic Coulomb phase (dash dotted curve) give similar results which are in satisfactory agreement with the experimental measurements of Schoffler et al. [25]. This is due to the absence of nuclear-nuclear coulomb distortion in MTCDW-4B model. However, the BDW-4B calculations underestimate the experimental data in general, specially at lower scattering angles. Around the critical angle $\theta_{{lab}}\thickapprox$ 0.24 mrad, a shoulder is seen in the dotted curve of BDW-4B model which is not observed in other theoretical works. It is a hybrid-type model which purposely coincides with the CDW-4B and the CB1 method in entrance and exit channels respectively. It may be mentioned that the BDW-4B calculation for DC have been performed by using one-parameter orbitals of the Hylleraas type [36] in both the entrance and exit channel. It may also be noted that the displayed theoretical and experimental results are only for the transition $1s^{2}\rightarrow{1s^{2}}$ . As can be seen from Fig. 5b, the BCCIS-4B theory (dash dotted curve) with asymptotic form gives results that agree closely with the data of Schoffler et al. [25] at small scattering angle (upto 0.27 mrad), but for large scattering angles the present results are smaller than the corresponding experimental measurement [25]. From Fig. 6, we observe the Thomas peak around the critical angle $\theta_{{lab}}\thickapprox$ 0.225 mrad both in the BCIS-4B [12] and the BDW-4B [27] calculations. Such structure is not observed in other theoretical calculations. However, our BCCIS-4B results are in good agreement with experimental results of Schuch et al. [40] above 0.3 mrad. In the small angular region, the present results (BCCIS-4B and TCDW-4B) overestimate the experimental data [40]. The curves of the BDW-4B and BCIS-4B are approximately coincident and underestimate the present and experimental results. The two sets of result (BDW-4B and BCIS-4B) agree within 20% in a very close vicinity of the extreme forward angle. In addition, there are two additional structures in the curve predicted by both the BCIS-4B and BDW-4B models, namely a minimum about $\theta_{{lab}}\approx$ 0.14 mrad and maximum about $\theta_{{lab}}\approx$ 0.22 mrad respectively. Above minimum angle $\theta_{{lab}}\approx$ 0.14 mrad, the BCIS-4B results is larger than the BDW-4B results by a factor of about 1.44. As can be seen from the Fig. 6, the CB1-4B approximation presents a sharp dip around 0.1 mrad which comes from the destructive interference of the partial amplitudes corresponding to the nuclear-electronic attractive and inter-nuclear repulsive interaction. This deep is unphysical, because it is not observed in the experimental measurements. Also, the other considered theoretical results do not exhibit such a behaviour to the collision process. The DCS in excited states for two electron transfer in $\textit{He}^{2+}-\textit{He}$ collisions are presented in Figs 7 and 8 at two different impact energies 60 keV/amu and 300 keV/amu. In Fig. 7 we observe some discrepancies between our present results (BCCIS-4B and MTCDW-4B) and the experimental data [44]. TC-BGM calculation in the framework of IEM shows excellent agreement with the experimental DCS for DC in excited state. However, at projectile energy above 300 keV/amu, all theoretical results show almost same and agreement with experiment is quite satisfactory.

4. Conclusions

The DC from helium atoms by high energy $\textit{He}^{2+}$ and $\textit{Li}^{3+}$ beams has been theoretically investigated by means of two quantum mechanical models, namely the BCCIS-4B and the MTCDW-4B. For ground state of He atom, the correlated multi-parameter wave functions have been used. We have found good agreement with the existing experimental data in particular when the BCCIS-4B model is used. For symmetric collisions ( $\textit{He}^{2+}-\textit{He}$ ), the contribution of TCS for DC from excited states is less important than the ground state capture when projectile energy increases. However, different features have been observed for asymmetric collision ( $\textit{Li}^{3+}-\textit{He}$ ). We have also investigated the DCS for $\textit{He}^{2+}-\textit{He}$ collision at different projectile energies in both ground and excited states. The obtained DCS by present models (BCCIS-4B and MTCDW-4B) are in good agreement with the available experimental data except at very small projectile scattering angle. Overall, we observe that the BCCIS-4B results show better agreement with the experiment than the MTCDW-4B results. From this investigation, we may find that any formulation on double electron capture may find some success only when: (i) proper boundary conditions for channel wavefunction and interactions are satisfied; (ii) continuum interaction of the active electrons is embodied; (iii) at least static correlation of the active electrons is properly taken into account and (iv) on-shell Coulomb wavefunction for any two-body Coulomb interaction is expected to be a better estimate than on-shell eikonal wavefunction.

Footnotes

Acknowledgments

The authors gratefully acknowledge the financial support from the Council of Scientific and Industrial Research (CSIR), New Delhi, India through project No: 03/1366/16/EMR-II.

Appendix A

Here $\sigma_{0}$ , $\sigma_{1}$ , $\sigma_{2}$ and $\sigma_{3}$ in Eq. (16a) and (16b) are given by

(A.1) $\displaystyle\sigma_{0}=a_{1}y^{2}+2y(\lambda_{2}a_{1}+a_{{11}}+a_{{12}})+e_{{% 11}}f_{{11}},\sigma_{1}=b_{1}y^{2}+2y(\lambda_{2}b_{1}+b_{{11}}+b_{{12}})+e_{{% 12}}f_{{11}},$ $\displaystyle\sigma_{2}=c_{1}y^{2}+2y(\lambda_{2}c_{1}+c_{{11}}+c_{{12}})+e_{{% 11}}f_{{12}},\sigma_{3}=d_{1}y^{2}+2y(\lambda_{2}d_{1}+d_{{11}}+d_{{12}})+e_{{% 12}}f_{{12}},$

where $a_{1}=|\vec{q}_{P}-\vec{q}|^{2}+\delta^{2}_{{11}}$ , $a_{{11}}=\delta_{2}(q^{2}_{P}+\delta^{2}_{{12}}+\lambda^{2}_{2})$ , $a_{{12}}=\delta_{{12}}P$ , $b_{1}=2\vec{v}_{f}.(\vec{q}_{P}-\vec{q})-2i\delta_{{11}}v_{f}$ , $b_{{11}}=-iv_{f}(q^{2}_{P}+\delta^{2}_{{12}}+\lambda^{2}_{2})$ , $b_{{12}}=-2\delta_{{12}}(\vec{q}.\vec{v}_{f}+i\delta_{2}v_{f})$ , $c_{1}=2\vec{k}_{f}.(\vec{q}_{P}-\vec{q})-2i\delta_{{11}}k_{f}$ , $c_{{11}}=2\delta_{2}\vec{q}_{P}.\vec{k}_{f}-2ik_{f}\delta_{{12}}\delta_{2}$ , $c_{{12}}=-ik_{f}P$ , $d_{1}=0$ , $d_{{11}}=2iv_{f}\vec{q}_{p}.\vec{k}_{f}-2\delta_{{12}}k_{f}v_{f}$ , $d_{{12}}=2i\vec{q}.\vec{v}_{f}k_{f}-2\delta_{2}v_{f}k_{f}$ , $e_{{11}}=q^{2}+\lambda^{2}_{{11}}$ , $e_{{12}}=-2\vec{q}.\vec{v}_{f}-2i\lambda_{{11}}v_{f}$ , $f_{{11}}=q^{2}_{P}+\delta^{2}_{{22}}$ , $f_{{12}}=2\vec{q}_{P}.\vec{k}_{f}-2ik_{f}\delta_{{22}}$ .

The terms of $\vec{q}_{P}$ , $\delta_{{11}}$ , $\delta_{{12}}$ , $P$ , $\lambda_{{11}}$ , $\Delta$ , $\vec{q}$ and $\delta_{{22}}$ , can be explicitly written as $\vec{q}_{P}=\vec{q}_{m}-(\vec{q}-t_{1}\vec{v}_{f})x$ , $\vec{q}_{m}=\vec{k}_{i}-\vec{k}_{f}$ , $\delta_{{11}}=(\delta_{2}+\Delta+\epsilon_{i})$ , $\delta_{{12}}=\Delta+\epsilon_{i},\epsilon_{i}\rightarrow 0$ , $P=q^{2}+\delta^{2}_{2}+\lambda^{2}_{2}$ , $\vec{q}=\frac{\vec{k}_{i}}{2+M_{T}}+\frac{\vec{k}_{f}}{2+M_{P}}$ , $\lambda_{{11}}=\delta_{2}+\lambda_{2}$ , $\delta_{{22}}=\Delta+\epsilon_{i}+\lambda_{2}$ , $\Delta^{2}=\beta^{2}_{1}x+\lambda^{2}_{1}(1-x)+x(1-x)(\vec{q}-t_{1}\vec{v}_{f}% )^{2}$ , $\beta_{1}=\delta_{1}-it_{1}v_{f}$ .

Here the terms $\delta_{1}$ , $\delta_{2}$ , $\lambda_{1}$ and $\lambda_{2}$ are the orbital component of the initial and final bound state wavefunctions. Using the one-parameter Hylleraas bound-state wavefunction in the entrance and exit channels only for the ground-to-ground state transition, the form of constant C is given by

$\displaystyle C=(\frac{{\delta_{1}}^{3}}{\pi})^{\frac{1}{2}}\left(\frac{{% \delta_{2}}^{3}}{\pi}\right)^{\frac{1}{2}}\left(\frac{{\lambda_{1}}^{3}}{\pi}% \right)^{\frac{1}{2}}\left(\frac{{\lambda_{2}}^{3}}{\pi}\right)^{\frac{1}{2}},$

where $\delta_{1}=\delta_{2}=\lambda_{1}=\lambda_{2}=$ 1.6875 for He atom of binding energies $\epsilon_{i}=\epsilon_{f}=$ $-$ 2.9037244 a.u. In Eq. (16a) we assume,

(A.2) $\displaystyle k={\sigma_{0}}^{i\alpha_{2}-i\alpha_{3}-1}(\sigma_{0}+\sigma_{1}% )^{-i\alpha_{2}}(\sigma_{0}+\sigma_{2})^{i\alpha_{3}}{{2}}F_{{1}}\{i\alpha_{{2% }};-i\alpha_{{3}};1;z\},$

where

$z=\frac{\sigma_{1}\sigma_{2}-\sigma_{0}\sigma_{3}}{(\sigma_{0}+\sigma_{1})(% \sigma_{0}+\sigma_{2})}.$

The first-order derivative of the Eq. (A.2) with respect to $\delta_{1}$ (say) in the parametric differential operator $D(\delta_{1},\delta_{2},\lambda_{1},\lambda_{2},\epsilon_{i})$ of Eq. (17) in the main text is given by

(A.3) $\displaystyle\frac{1}{k}\frac{\partial{k}}{\partial{\delta_{1}}}=(i\alpha_{2}-% i\alpha_{3}-1)\frac{\partial{\sigma_{0}}}{\partial{\delta_{1}}}+(-i\alpha_{2})% \left(\frac{\partial{\sigma_{0}}}{\partial{\delta_{1}}}+\frac{\partial{\sigma_% {1}}}{\partial{\delta_{1}}}\right)+(i\alpha_{3})\left(\frac{\partial{\sigma_{0% }}}{\partial{\delta_{1}}}+\frac{\partial{\sigma_{2}}}{\partial{\delta_{1}}}% \right)+\frac{{{2}}F_{{1}}(i\alpha_{{2}}+1;-i\alpha_{{3}}+1;2;z)}{{{2}}F_{{1}}% (i\alpha_{{2}};-i\alpha_{{3}};1;z)}\frac{\partial{z}}{\partial{\delta_{1}}},$

where the first-order derivatives of $\sigma_{0}$ , $\sigma_{1}$ , $\sigma_{2}$ and $\sigma_{3}$ can be written as

$\displaystyle\frac{\partial{\sigma_{0}}}{\partial{\delta_{1}}}=Q_{{01}}\frac{% \partial{\bigtriangleup}}{\partial{\delta_{1}}},\frac{\partial{\sigma_{1}}}{% \partial{\delta_{1}}}=Q_{{11}}\frac{\partial{\bigtriangleup}}{\partial{\delta_% {1}}},\frac{\partial{\sigma_{2}}}{\partial{\delta_{1}}}=Q_{{21}}\frac{\partial% {\bigtriangleup}}{\partial{\delta_{1}}},\frac{\partial{\sigma_{3}}}{\partial{% \delta_{1}}}=Q_{{31}}\frac{\partial{\bigtriangleup}}{\partial{\delta_{1}}},$

with

$\displaystyle Q_{{01}}=2\delta_{{11}}y^{2}+2y(2\lambda_{2}\delta_{{11}}+2% \delta_{2}\delta_{{12}}+P)+2{e_{{11}}}\delta_{{22}},\frac{\partial{% \bigtriangleup}}{\partial{\delta_{1}}}=\frac{(\delta_{1}-i{v_{f}}t_{1})x}{% \bigtriangleup},$ $\displaystyle Q_{{11}}=-2i{v_{f}}y^{2}+2y\{-2i{v_{f}}(\delta_{{22}}+\delta_{2}% )-2{\vec{q}.{\vec{v}}_{f}}\}+2{e_{{12}}}\delta_{{22}},Q_{{21}}=-2i{k_{f}}(y^{2% }+2y\lambda_{{11}}+e_{{11}}),$ $\displaystyle Q_{{31}}=-2{k_{f}}(2yv_{f}+ie_{{12}}).$

Similarly we can perform the partial derivatives of k (Eq. (A.2)) with respect to $\delta_{2}$ , $\lambda_{1}$ , $\lambda_{2}$ and $\epsilon_{i}$ in the differential operator $D(\delta_{1},\delta_{2},\lambda_{1},\lambda_{2},\epsilon_{i})$ in Eq. (17) of the main text which becomes

$\displaystyle\frac{\partial{\sigma_{0}}}{\partial{\delta_{2}}}=2\delta_{{11}}y% ^{2}+2y(\lambda_{2}\delta_{{11}}+{q_{p}}^{2}+{\delta_{{12}}}^{2}+{\lambda_{2}}% ^{2}+2\delta_{2}\delta_{{12}})+2\lambda_{{11}}f_{{11}},\frac{\partial{\sigma_{% 1}}}{\partial{\delta_{2}}}=-2iv_{f}(y^{2}+2y\delta_{{22}}+f_{{11}}),$ $\displaystyle\frac{\partial{\sigma_{2}}}{\partial{\delta_{2}}}=-2ik_{f}\{y^{2}% +2y(\lambda_{{11}}+\delta_{{12}})\}+4y{\vec{q}_{P}}.{\vec{k}_{f}}+2\lambda_{{1% 1}}f_{{12}},\frac{\partial{\sigma_{3}}}{\partial{\delta_{2}}}=-4yk_{f}v_{f}-2% iv_{f}f_{{12}},$ $\displaystyle\frac{\partial{\sigma_{0}}}{\partial{\lambda_{1}}}=Q_{{01}}\frac{% \partial{\bigtriangleup}}{\partial{\lambda_{1}}},\ \text{where}\ \frac{% \partial{\bigtriangleup}}{\partial{\lambda_{1}}}=\frac{(1-x)\lambda_{1}}{% \bigtriangleup}.$ $\displaystyle\frac{\partial{\sigma_{1}}}{\partial{\lambda_{1}}}=Q_{{11}}\frac{% \partial{\bigtriangleup}}{\partial{\lambda_{1}}},\frac{\partial{\sigma_{2}}}{% \partial{\lambda_{1}}}=Q_{{21}}\frac{\partial{\bigtriangleup}}{\partial{% \lambda_{1}}},\frac{\partial{\sigma_{3}}}{\partial{\lambda_{1}}}=Q_{{31}}\frac% {\partial{\bigtriangleup}}{\partial{\lambda_{1}}},$ $\displaystyle\frac{\partial{\sigma_{0}}}{\partial{\lambda_{2}}}=2y(a_{1}+2% \lambda_{2}\delta_{{11}})+2\lambda_{{11}}f_{{11}}+2\delta_{{22}}e_{{11}},\frac% {\partial{\sigma_{1}}}{\partial{\lambda_{2}}}=2y(a_{1}-2iv_{f}\lambda_{2})+2% \delta_{{22}}e_{{12}}-2iv_{f}f_{{11}},$ $\displaystyle\frac{\partial{\sigma_{2}}}{\partial{\lambda_{2}}}=2y(a_{1}-2ik_{% f}\lambda_{2})+2(\delta_{2}+\lambda_{2})f_{{12}}-2ik_{f}e_{{11}},\frac{% \partial{\sigma_{3}}}{\partial{\lambda_{2}}}=2ya_{1}-2iv_{f}f_{{12}}-2ik_{f}e_% {{12}},$ $\displaystyle\frac{\partial{\sigma_{0}}}{\partial{\epsilon_{i}}}=Q_{{01}},% \frac{\partial{\sigma_{1}}}{\partial{\epsilon_{i}}}=Q_{{11}},\frac{\partial{% \sigma_{2}}}{\partial{\epsilon_{i}}}=Q_{{21}},\frac{\partial{\sigma_{3}}}{% \partial{\epsilon_{i}}}=Q_{{31}}.$

Using forth-order derivatives of Eq. (A.2) with respect to the variables $(\delta_{1},\delta_{2},\lambda_{1},\lambda_{2},\epsilon_{i})$ and differential operator $D(\delta_{1},\delta_{2},\lambda_{1},\lambda_{2},\epsilon_{i})$ in Eq. (17) of the main text, we can calculate the ground-to-ground state transition by the higher order derivatives of $\sigma_{0}$ , $\sigma_{1}$ , $\sigma_{2}$ , $\sigma_{3}$ which becomes

$\displaystyle\frac{{\partial^{2}}{\sigma_{0}}}{\partial{\delta_{1}}\partial{% \delta_{2}}}=2\{y^{2}+2y(\delta_{{22}}+\delta_{2})+\lambda_{{11}}\delta_{{22}}% \}\frac{\partial{\bigtriangleup}}{\partial{\delta_{1}}},\frac{{\partial^{2}}{% \sigma_{1}}}{\partial{\delta_{1}}\partial{\delta_{2}}}=-4iv_{f}(y+\delta_{{22}% })\frac{\partial{\bigtriangleup}}{\partial{\delta_{1}}},$ $\displaystyle\frac{{\partial^{2}}{\sigma_{2}}}{\partial{\delta_{1}}\partial{% \delta_{2}}}=-4ik_{f}(y+\lambda_{{11}})\frac{\partial{\bigtriangleup}}{% \partial{\delta_{1}}},\frac{{\partial^{2}}{\sigma_{3}}}{\partial{\delta_{1}}% \partial{\delta_{2}}}=-4v_{f}k_{f}\frac{\partial{\bigtriangleup}}{\partial{% \delta_{1}}},\frac{{\partial^{2}}{\sigma_{0}}}{\partial{\delta_{1}}\partial{% \lambda_{1}}}=Q_{{01}}\frac{{\partial^{2}}{\bigtriangleup}}{\partial{\delta_{1% }}\partial{\lambda_{1}}}+2{P_{{01}}}\frac{\partial{\bigtriangleup}}{\partial{% \lambda_{1}}}\frac{\partial{\bigtriangleup}}{\partial{\delta_{1}}},$

where

$\displaystyle P_{{01}}=y^{2}+2y\lambda_{{11}}+e_{{11}},\frac{{\partial^{2}}{% \bigtriangleup}}{\partial{\delta_{1}}\partial{\lambda_{1}}}=-\frac{x(1-x)% \lambda_{1}(\delta_{1}-iv_{f}t_{1})}{\bigtriangleup^{3}}.$ $\displaystyle\frac{{\partial^{2}}{\sigma_{1}}}{\partial{\delta_{1}}\partial{% \lambda_{1}}}=Q_{{11}}\frac{{\partial^{2}}{\bigtriangleup}}{\partial{\delta_{1% }}\partial{\lambda_{1}}}+2(e_{{12}}-2iyv_{f})\frac{\partial{\bigtriangleup}}{% \partial{\lambda_{1}}}\frac{\partial{\bigtriangleup}}{\partial{\delta_{1}}},% \frac{{\partial^{2}}{\sigma_{2}}}{\partial{\delta_{1}}\partial{\lambda_{1}}}=Q% _{{21}}\frac{{\partial^{2}}{\bigtriangleup}}{\partial{\delta_{1}}\partial{% \lambda_{1}}},\frac{{\partial^{2}}{\sigma_{3}}}{\partial{\delta_{1}}\partial{% \lambda_{1}}}=$ $\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Q_{{31}}\frac{{\partial^{2}% }{\bigtriangleup}}{\partial{\delta_{1}}\partial{\lambda_{1}}},$ $\displaystyle\frac{{\partial^{3}}{\sigma_{0}}}{\partial{\delta_{1}}\partial{% \delta_{2}}\partial{\lambda_{1}}}=2\{y^{2}+2y(\lambda_{{11}}+\delta_{{12}})+2% \lambda_{{11}}\delta_{{22}}\}\frac{{\partial^{2}}{\bigtriangleup}}{\partial{% \delta_{1}}\partial{\lambda_{1}}}+2(y+\lambda_{{11}})\frac{\partial{% \bigtriangleup}}{\partial{\lambda_{1}}}\frac{\partial{\bigtriangleup}}{% \partial{\delta_{1}}},$ $\displaystyle\frac{{\partial^{4}}{\sigma_{0}}}{\partial{\delta_{1}}\partial{% \delta_{2}}\partial{\lambda_{1}}\partial{\lambda_{2}}}=2\{y^{2}+2(y+\delta_{{2% 2}}+\lambda_{{11}})\}\frac{{\partial^{2}}{\bigtriangleup}}{\partial{\delta_{1}% }\partial{\lambda_{1}}}+2\frac{\partial{\bigtriangleup}}{\partial{\lambda_{1}}% }\frac{\partial{\bigtriangleup}}{\partial{\delta_{1}}},\frac{{\partial^{3}}{% \sigma_{1}}}{\partial{\delta_{1}}\partial{\lambda_{1}}\partial{\delta_{2}}}$ $\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =-4iv_{f}(y% +\delta_{{22}})\frac{{\partial^{2}}{\bigtriangleup}}{\partial{\delta_{1}}% \partial{\lambda_{1}}}-4iv_{f}\frac{\partial{\bigtriangleup}}{\partial{\lambda% _{1}}}\frac{\partial{\bigtriangleup}}{\partial{\delta_{1}}},$ $\displaystyle\frac{{\partial^{4}}{\sigma_{1}}}{\partial{\delta_{1}}\partial{% \delta_{2}}\partial{\lambda_{1}}\partial{\lambda_{2}}}=-4iv_{f}\frac{{\partial% ^{2}}{\bigtriangleup}}{\partial{\lambda_{1}}\partial{\delta_{1}}},\frac{{% \partial^{3}}{\sigma_{2}}}{\partial{\delta_{1}}\partial{\lambda_{1}}\partial{% \delta_{2}}}=-4ik_{f}(y+\lambda_{{11}})\frac{{\partial^{2}}{\bigtriangleup}}{% \partial{\delta_{1}}\partial{\lambda_{1}}},\frac{{\partial^{4}}{\sigma_{2}}}{% \partial{\delta_{1}}\partial{\lambda_{1}}\partial{\delta_{2}}\partial{\lambda_% {2}}}$ $\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =-4ik_{f}% \frac{{\partial^{2}}{\bigtriangleup}}{\partial{\delta_{1}}\partial{\lambda_{1}% }},$ $\displaystyle\frac{{\partial^{3}}{\sigma_{3}}}{\partial{\delta_{1}}\partial{% \lambda_{1}}\partial{\delta_{2}}}=-4k_{f}v_{f}\frac{{\partial^{2}}{% \bigtriangleup}}{\partial{\delta_{1}}\partial{\lambda_{1}}},\frac{{\partial^{4% }}{\sigma_{3}}}{\partial{\delta_{1}}\partial{\lambda_{1}}\partial{\delta_{2}}% \partial{\lambda_{2}}}=0,\frac{{\partial^{4}}{\sigma_{0}}}{\partial{\delta_{1}% }\partial{\delta_{2}}\partial{\lambda_{1}}\partial{\epsilon_{i}}}$ $\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =4(y+% \lambda_{{11}})\frac{{\partial^{2}}{\bigtriangleup}}{\partial{\delta_{1}}% \partial{\lambda_{1}}},\frac{{\partial^{4}}{\sigma_{1}}}{\partial{\delta_{1}}% \partial{\delta_{2}}\partial{\lambda_{1}}\partial{\epsilon_{i}}}=-4iv_{f}\frac% {{\partial^{2}}{\bigtriangleup}}{\partial{\delta_{1}}\partial{\lambda_{1}}},$ $\displaystyle\frac{{\partial^{4}}{\sigma_{2}}}{\partial{\delta_{1}}\partial{% \lambda_{1}}\partial{\delta_{2}}\partial{\epsilon_{i}}}=0,\frac{{\partial^{4}}% {\sigma_{3}}}{\partial{\delta_{1}}\partial{\delta_{2}}\partial{\lambda_{1}}% \partial{\epsilon_{i}}}=0,\frac{{\partial^{3}}{\sigma_{0}}}{\partial{\delta_{1% }}\partial{\delta_{2}}\partial{\lambda_{2}}}=2(2y+\delta_{{22}}+\lambda_{{11}}% )\frac{\partial{\bigtriangleup}}{\partial{\delta_{1}}},\frac{{\partial^{4}}{% \sigma_{0}}}{\partial{\delta_{1}}\partial{\delta_{2}}\partial{\lambda_{2}}% \partial{\epsilon_{i}}}$ $\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =2\frac{% \partial{\bigtriangleup}}{\partial{\delta_{1}}},$ $\displaystyle\frac{{\partial^{3}}{\sigma_{1}}}{\partial{\delta_{1}}\partial{% \delta_{2}}\partial{\lambda_{2}}}=-4iv_{f}\frac{\partial{\bigtriangleup}}{% \partial{\delta_{1}}},\frac{{\partial^{4}}{\sigma_{1}}}{\partial{\delta_{1}}% \partial{\delta_{2}}\partial{\lambda_{2}}\partial{\epsilon_{i}}}=0,\frac{{% \partial^{3}}{\sigma_{2}}}{\partial{\delta_{1}}\partial{\delta_{2}}\partial{% \lambda_{2}}}=-4ik_{f}\frac{\partial{\bigtriangleup}}{\partial{\delta_{1}}},% \frac{{\partial^{4}}{\sigma_{2}}}{\partial{\delta_{1}}\partial{\delta_{2}}% \partial{\lambda_{2}}\partial{\epsilon_{i}}}=0,$ $\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{{% \partial^{3}}{\sigma_{3}}}{\partial{\delta_{1}}\partial{\delta_{2}}\partial{% \lambda_{2}}}=0,\frac{{\partial^{4}}{\sigma_{3}}}{\partial{\delta_{1}}\partial% {\delta_{2}}\partial{\lambda_{2}}\partial{\epsilon_{i}}}=0 .$

Appendix B

Here the form of $\sigma_{0}$ , $\sigma_{1}$ , $\sigma_{2}$ and $\sigma_{3}$ are similar in Appendix A (Eq. (A.1)), where

$\displaystyle a_{1}=q^{2}+\delta^{2}_{{11}},a_{{11}}=\delta_{2}(q^{2}+\delta^{% 2}_{{12}}+\lambda^{2}_{1}),a_{{12}}=\delta_{{12}}(\delta^{2}_{2}+\lambda^{2}_{% 2}),b_{1}=2\vec{q}.{\vec{v}}_{f}-2iv_{f}\delta_{{11}},$ $\displaystyle b_{{11}}=-iv_{f}(q^{2}+\lambda^{2}_{2}+\delta^{2}_{{12}}),b_{{12% }}=-2iv_{f}\delta_{2}\delta_{{12}},c_{1}=2\vec{q}.{\vec{k}}_{f}-2ik_{f}\delta_% {{11}},c_{{11}}=2\vec{q}.{\vec{k}}_{f}\delta_{2}-2ik_{f}\delta_{2}\delta_{{12}},$ $\displaystyle c_{{12}}=-ik_{f}(\delta^{2}_{2}+\lambda^{2}_{2}),d_{1}=0,d_{{11}% }=-2iv_{f}\vec{q}.{\vec{k}}_{f}-2k_{f}v_{f}\delta_{{12}},d_{{12}}=-2\delta_{2}% k_{f}v_{f},e_{{11}}=\lambda^{2}_{{11}},$ $\displaystyle e_{{12}}=-2iv_{f}\lambda_{{11}},f_{{11}}=q^{2}+\delta^{2}_{{22}}% ,f_{{12}}=2\vec{q}.{\vec{k}}_{f}-2ik_{f}\delta_{{22}},$

with

$\displaystyle\delta_{{11}}=\delta_{2}+\bigtriangleup+\epsilon_{i},\delta_{{12}% }=\bigtriangleup+\epsilon_{i},\lambda_{{11}}=\delta_{2}+\lambda_{2},\delta_{{2% 2}}=\bigtriangleup+\epsilon_{i}+\lambda_{2},\vec{q}=\vec{k}_{i}-\vec{k}_{f},$ $\displaystyle\bigtriangleup^{2}=x(1-x)v^{2}_{f}t^{2}_{1}+(\delta_{1}-it_{1}v_{% f})^{2}x+(1-x)\lambda^{2}_{1}.$

We perform the successive derivative up to fourth order of k (equation (A2)) for ground-to-ground state transition using the explicit form of the following derivatives:

$\displaystyle\frac{\partial{\sigma_{0}}}{\partial{\delta_{1}}}=\{2\delta_{{11}% }y^{2}+2y(2\delta_{{11}}\lambda_{2}+2\delta_{2}\delta_{{12}}+\delta^{2}_{2}+% \lambda^{2}_{2})+2e_{{11}}\delta_{{22}}\}\frac{\partial{\bigtriangleup}}{% \partial{\delta_{1}}},$ $\displaystyle\frac{{\partial^{2}}{\sigma_{0}}}{\partial{\delta_{1}}\partial{% \delta_{2}}}=\{2y^{2}+2y(2\lambda_{2}+2\delta_{{12}}+2\delta_{2})+4\lambda_{{1% 1}}\delta_{{22}}\}\frac{\partial{\bigtriangleup}}{\partial{\delta_{1}}},$ $\displaystyle\frac{{\partial^{3}}{\sigma_{0}}}{\partial{\delta_{1}}\partial{% \delta_{2}}\partial{\lambda_{1}}}=4\lambda_{{11}}\frac{\partial{\bigtriangleup% }}{\partial{\lambda_{1}}}\frac{\partial{\bigtriangleup}}{\partial{\delta_{1}}}% +\{2y^{2}+4y(\delta_{{12}}+\lambda_{{11}})+4\lambda_{{11}}\delta_{{22}}\}\frac% {{\partial^{2}\bigtriangleup}}{\partial{\delta_{1}}\partial{\lambda_{1}}},$ $\displaystyle\frac{{\partial^{4}}{\sigma_{0}}}{\partial{\delta_{1}}\partial{% \delta_{2}}\partial{\lambda_{1}}\partial{\lambda_{2}}}=4\frac{\partial{% \bigtriangleup}}{\partial{\lambda_{1}}}\frac{\partial{\bigtriangleup}}{% \partial{\delta_{1}}}+4(y+\delta_{{22}}+\lambda_{{11}})\frac{{\partial^{2}% \bigtriangleup}}{\partial{\delta_{1}}\partial{\lambda_{1}}},\frac{{\partial^{4% }}{\sigma_{0}}}{\partial{\delta_{1}}\partial{\delta_{2}}\partial{\lambda_{1}}% \partial{\epsilon_{i}}}=4(y+\lambda_{{11}})\frac{{\partial^{2}\bigtriangleup}}% {\partial{\delta_{1}}\partial{\lambda_{1}}},$ $\displaystyle\frac{{\partial^{3}}{\sigma_{0}}}{\partial{\delta_{1}}\partial{% \delta_{2}}\partial{\lambda_{2}}}=4(y+\delta_{{22}}+\lambda_{{11}})\frac{% \partial{\bigtriangleup}}{\partial{\delta_{1}}},\frac{{\partial^{4}}{\sigma_{0% }}}{\partial{\delta_{1}}\partial{\delta_{2}}\partial{\lambda_{2}}\partial{% \epsilon_{i}}}=4\frac{\partial{\bigtriangleup}}{\partial{\delta_{1}}},$ $\displaystyle\frac{\partial{\sigma_{1}}}{\partial{\delta_{1}}}=-2iv_{f}\{y^{2}% +2y(\lambda_{{11}}+\delta_{{12}})\}\frac{\partial{\bigtriangleup}}{\partial{% \delta_{1}}}+2\delta_{{22}}e_{{12}}\frac{\partial{\bigtriangleup}}{\partial{% \delta_{1}}},\frac{{\partial^{2}}{\sigma_{1}}}{\partial{\delta_{1}}\partial{% \delta_{2}}}=-4iv_{f}(y+\delta_{{22}})\frac{\partial{\bigtriangleup}}{\partial% {\delta_{1}}},$ $\displaystyle\frac{{\partial^{3}}{\sigma_{1}}}{\partial{\delta_{1}}\partial{% \delta_{2}}\partial{\lambda_{1}}}=-4iv_{f}(y+\delta_{{22}})\frac{{\partial^{2}% }{\bigtriangleup}}{\partial{\delta_{1}}\partial{\lambda_{1}}}-4iv_{f}\frac{% \partial{\bigtriangleup}}{\partial{\delta_{1}}}\frac{\partial{\bigtriangleup}}% {\partial{\lambda_{1}}},\frac{{\partial^{4}}{\sigma_{1}}}{\partial{\delta_{1}}% \partial{\delta_{2}}\partial{\lambda_{1}}\partial{\lambda_{2}}}=-4iv_{f}\frac{% {\partial^{2}}{\bigtriangleup}}{\partial{\delta_{1}}\partial{\lambda_{1}}},$ $\displaystyle\frac{{\partial^{4}}{\sigma_{1}}}{\partial{\delta_{1}}\partial{% \delta_{2}}\partial{\lambda_{1}}\partial{\epsilon_{i}}}=-4iv_{f}\frac{{% \partial^{2}}{\bigtriangleup}}{\partial{\delta_{1}}\partial{\lambda_{1}}},% \frac{{\partial^{3}}{\sigma_{1}}}{\partial{\delta_{1}}\partial{\delta_{2}}% \partial{\lambda_{2}}}=-4iv_{f}\frac{\partial{\bigtriangleup}}{\partial{\delta% _{1}}},\frac{{\partial^{4}}{\sigma_{1}}}{\partial{\delta_{1}}\partial{\delta_{% 2}}\partial{\lambda_{2}}\partial{\epsilon_{i}}}=0,$ $\displaystyle\frac{\partial{\sigma_{2}}}{\partial{\delta_{1}}}=-2ik_{f}(y^{2}+% 2y\lambda_{{11}}+e_{{11}})\frac{\partial{\bigtriangleup}}{\partial{\delta_{1}}% },\frac{{\partial^{2}}{\sigma_{2}}}{\partial{\delta_{1}}\partial{\delta_{2}}}=% -4ik_{f}(y+\lambda_{{11}})\frac{\partial{\bigtriangleup}}{\partial{\delta_{1}}% },\frac{{\partial^{3}}{\sigma_{2}}}{\partial{\delta_{1}}\partial{\delta_{2}}% \partial{\lambda_{1}}}$ $\displaystyle\ \ \ \ \ \ \ \ \ =-4ik_{f}(y+\lambda_{{11}})\frac{{\partial^{2}}% {\bigtriangleup}}{\partial{\delta_{1}}\partial{\lambda_{1}}},$ $\displaystyle\frac{{\partial^{4}}{\sigma_{2}}}{\partial{\delta_{1}}\partial{% \delta_{2}}\partial{\lambda_{1}}\partial{\lambda_{2}}}=-4ik_{f}\frac{{\partial% ^{2}}{\bigtriangleup}}{\partial{\delta_{1}}\partial{\lambda_{1}}},\frac{{% \partial^{4}}{\sigma_{2}}}{\partial{\delta_{1}}\partial{\delta_{2}}\partial{% \lambda_{1}}\partial{\epsilon_{i}}}=0,\frac{{\partial^{3}}{\sigma_{2}}}{% \partial{\delta_{1}}\partial{\delta_{2}}\partial{\lambda_{2}}}=-4ik_{f}\frac{% \partial{\bigtriangleup}}{\partial{\delta_{1}}},$ $\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ % \frac{{\partial^{4}}{\sigma_{2}}}{\partial{\delta_{1}}\partial{\delta_{2}}% \partial{\lambda_{2}}\partial{\epsilon_{i}}}=0,$ $\displaystyle\frac{\partial{\sigma_{3}}}{\partial{\delta_{1}}}=-4yk_{f}v_{f}% \frac{\partial{\bigtriangleup}}{\partial{\delta_{1}}}-2ik_{f}e_{{12}}\frac{% \partial{\bigtriangleup}}{\partial{\delta_{1}}},\frac{{\partial^{2}}{\sigma_{3% }}}{\partial{\delta_{1}}\partial{\delta_{2}}}=-4k_{f}v_{f}\frac{\partial{% \bigtriangleup}}{\partial{\delta_{1}}},\frac{{\partial^{3}}{\sigma_{3}}}{% \partial{\delta_{1}}\partial{\delta_{2}}\partial{\lambda_{1}}}=-4k_{f}v_{f}% \frac{{\partial^{2}}{\bigtriangleup}}{\partial{\delta_{1}}\partial{\lambda_{1}% }},$ $\displaystyle\frac{{\partial^{4}}{\sigma_{3}}}{\partial{\delta_{1}}\partial{% \delta_{2}}\partial{\lambda_{1}}\partial{\epsilon_{i}}}=0,\frac{{\partial^{3}}% {\sigma_{3}}}{\partial{\delta_{1}}\partial{\delta_{2}}\partial{\lambda_{2}}}=0% ,\frac{{\partial^{4}}{\sigma_{3}}}{\partial{\delta_{1}}\partial{\delta_{2}}% \partial{\lambda_{1}}\partial{\lambda_{2}}}=0,\frac{{\partial^{4}}{\sigma_{3}}% }{\partial{\delta_{1}}\partial{\delta_{2}}\partial{\lambda_{2}}\partial{% \epsilon_{i}}}=0.$

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Double electron capture in fast ion-atom collisions

Abstract

Keywords

1. Introduction

2. Theory

Table 1 CS (in c ⁢ m 2 ) for ground-to-ground state DC (using the one-parameter helium wavefunction of Hylleraas [36] in the initial and final state) as a function of the incident energy (keV) for the collision 𝐻𝑒 2 + - 𝐻𝑒

Table 2 TCS (in c ⁢ m 2 ) for DC into both ground and excited states as a function of the incident energy (keV) for the collision 𝐻𝑒 2 + - 𝐻𝑒

Table 3 DCS ( c ⁢ m 2 / s ⁢ r ) into the ground state as a function of the scattering angle θ (mrad) at the incident energy 150 keV/amu for the DC cross sections in 𝐻𝑒 2 + - 𝐻𝑒 ⁢ ( 1 ⁢ s 2 ) collisions

Footnotes

Acknowledgments

Appendix A

Appendix B

References

Table 1
CS (in $cm^{2}$ ) for ground-to-ground state DC (using the one-parameter helium wavefunction of Hylleraas [36] in the initial and final state) as a function of the incident energy (keV) for the collision $\textit{He}^{2+}-\textit{He}$

Table 2
TCS (in $cm^{2}$ ) for DC into both ground and excited states as a function of the incident energy (keV) for the collision $\textit{He}^{2+}-\textit{He}$

Table 3
DCS ( $cm^{2}/sr$ ) into the ground state as a function of the scattering angle $\theta$ (mrad) at the incident energy 150 keV/amu for the DC cross sections in $\textit{He}^{2+}-\textit{He}(1s^{2})$ collisions